Yunus Emre Demirci , Tınaz Ekim , John Gimbel , Mehmet Akif Yıldız
{"title":"图类中有缺陷Ramsey数的精确值","authors":"Yunus Emre Demirci , Tınaz Ekim , John Gimbel , Mehmet Akif Yıldız","doi":"10.1016/j.disopt.2021.100673","DOIUrl":null,"url":null,"abstract":"<div><p>Given a graph <span><math><mi>G</mi></math></span>, a <span><math><mi>k</mi></math></span><em>-sparse</em> <span><math><mi>j</mi></math></span><em>-set</em> is a set of <span><math><mi>j</mi></math></span><span> vertices inducing a subgraph with maximum degree at most </span><span><math><mi>k</mi></math></span>. A <span><math><mi>k</mi></math></span><em>-dense</em> <span><math><mi>i</mi></math></span><em>-set</em> is a set of <span><math><mi>i</mi></math></span> vertices that is <span><math><mi>k</mi></math></span>-sparse in the complement of <span><math><mi>G</mi></math></span>. As a generalization of Ramsey numbers, the <span><math><mi>k</mi></math></span>-defective Ramsey number <span><math><mrow><msubsup><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>G</mi></mrow></msubsup><mrow><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo></mrow></mrow></math></span> for the graph class <span><math><mi>G</mi></math></span> is defined as the smallest natural number <span><math><mi>n</mi></math></span> such that all graphs on <span><math><mi>n</mi></math></span> vertices in the class <span><math><mi>G</mi></math></span> have either a <span><math><mi>k</mi></math></span>-dense <span><math><mi>i</mi></math></span>-set or a <span><math><mi>k</mi></math></span>-sparse <span><math><mi>j</mi></math></span>-set. In this paper, we examine <span><math><mrow><msubsup><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>G</mi></mrow></msubsup><mrow><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo></mrow></mrow></math></span> where <span><math><mi>G</mi></math></span><span> represents various graph classes. For forests and cographs, we give exact formulas for all defective Ramsey numbers. For cacti, bipartite graphs and split graphs, we derive defective Ramsey numbers in most of the cases and point out open questions, formulated as conjectures if possible.</span></p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Exact values of defective Ramsey numbers in graph classes\",\"authors\":\"Yunus Emre Demirci , Tınaz Ekim , John Gimbel , Mehmet Akif Yıldız\",\"doi\":\"10.1016/j.disopt.2021.100673\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Given a graph <span><math><mi>G</mi></math></span>, a <span><math><mi>k</mi></math></span><em>-sparse</em> <span><math><mi>j</mi></math></span><em>-set</em> is a set of <span><math><mi>j</mi></math></span><span> vertices inducing a subgraph with maximum degree at most </span><span><math><mi>k</mi></math></span>. A <span><math><mi>k</mi></math></span><em>-dense</em> <span><math><mi>i</mi></math></span><em>-set</em> is a set of <span><math><mi>i</mi></math></span> vertices that is <span><math><mi>k</mi></math></span>-sparse in the complement of <span><math><mi>G</mi></math></span>. As a generalization of Ramsey numbers, the <span><math><mi>k</mi></math></span>-defective Ramsey number <span><math><mrow><msubsup><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>G</mi></mrow></msubsup><mrow><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo></mrow></mrow></math></span> for the graph class <span><math><mi>G</mi></math></span> is defined as the smallest natural number <span><math><mi>n</mi></math></span> such that all graphs on <span><math><mi>n</mi></math></span> vertices in the class <span><math><mi>G</mi></math></span> have either a <span><math><mi>k</mi></math></span>-dense <span><math><mi>i</mi></math></span>-set or a <span><math><mi>k</mi></math></span>-sparse <span><math><mi>j</mi></math></span>-set. In this paper, we examine <span><math><mrow><msubsup><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>G</mi></mrow></msubsup><mrow><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo></mrow></mrow></math></span> where <span><math><mi>G</mi></math></span><span> represents various graph classes. For forests and cographs, we give exact formulas for all defective Ramsey numbers. For cacti, bipartite graphs and split graphs, we derive defective Ramsey numbers in most of the cases and point out open questions, formulated as conjectures if possible.</span></p></div>\",\"PeriodicalId\":50571,\"journal\":{\"name\":\"Discrete Optimization\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2021-11-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discrete Optimization\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S1572528621000529\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Optimization","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1572528621000529","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Exact values of defective Ramsey numbers in graph classes
Given a graph , a -sparse-set is a set of vertices inducing a subgraph with maximum degree at most . A -dense-set is a set of vertices that is -sparse in the complement of . As a generalization of Ramsey numbers, the -defective Ramsey number for the graph class is defined as the smallest natural number such that all graphs on vertices in the class have either a -dense -set or a -sparse -set. In this paper, we examine where represents various graph classes. For forests and cographs, we give exact formulas for all defective Ramsey numbers. For cacti, bipartite graphs and split graphs, we derive defective Ramsey numbers in most of the cases and point out open questions, formulated as conjectures if possible.
期刊介绍:
Discrete Optimization publishes research papers on the mathematical, computational and applied aspects of all areas of integer programming and combinatorial optimization. In addition to reports on mathematical results pertinent to discrete optimization, the journal welcomes submissions on algorithmic developments, computational experiments, and novel applications (in particular, large-scale and real-time applications). The journal also publishes clearly labelled surveys, reviews, short notes, and open problems. Manuscripts submitted for possible publication to Discrete Optimization should report on original research, should not have been previously published, and should not be under consideration for publication by any other journal.